On the Relations among Characteristic Functions of Theta Functions
Abstract
In this study, using the characteristic values $\begin{bmatrix} \varepsilon\\ \varepsilon' \end{bmatrix} = \begin{bmatrix} 1\\ 1 \end{bmatrix}, \begin{bmatrix} 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0 \end{bmatrix} \pmod 2$ a theorem on the $\frac{1}{2^r}$ coefficients of periods of first order theta function according to the $(1,τ)$ period pair (for $r \in N^+$) is established. The following equalities are also obtained.
- $\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{1}{2^r}-\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$
- $\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{\pi i}{2^r}}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$
DOI: https://doi.org/10.3844/jmssp.2005.142.145
Copyright: © 2005 Ismet Yildiz and Neslihan Uyanik. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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Keywords
- First Order Theta Function
- Characteristic Values